Effective chiral magnetic currents, topological magnetic charges, and microwave vortices in a cavity with an enclosed ferrite disk
📝 Abstract
In microwaves, a TE-polarized rectangular-waveguide resonator with an inserted thin ferrite disk gives an example of a nonintegrable system. The interplay of reflection and transmission at the disk interfaces together with the material gyrotropy effect gives rise to whirlpool-like electromagnetic vortices in the proximity of the ferromagnetic resonance. Based on numerical simulation, we show that a character of microwave vortices in a cavity can be analyzed by means of consideration of equivalent magnetic currents. Maxwell equations allows introduction of a magnetic current as a source of the electromagnetic field. Specifically, we found that in such nonintegrable structures, magnetic gyrotropy and geometrical factors leads to the effect of symmetry breaking resulting in effective chiral magnetic currents and topological magnetic charges. As an intriguing fact, one can observe precessing behavior of the electric-dipole polarization inside a ferrite disk.
💡 Analysis
In microwaves, a TE-polarized rectangular-waveguide resonator with an inserted thin ferrite disk gives an example of a nonintegrable system. The interplay of reflection and transmission at the disk interfaces together with the material gyrotropy effect gives rise to whirlpool-like electromagnetic vortices in the proximity of the ferromagnetic resonance. Based on numerical simulation, we show that a character of microwave vortices in a cavity can be analyzed by means of consideration of equivalent magnetic currents. Maxwell equations allows introduction of a magnetic current as a source of the electromagnetic field. Specifically, we found that in such nonintegrable structures, magnetic gyrotropy and geometrical factors leads to the effect of symmetry breaking resulting in effective chiral magnetic currents and topological magnetic charges. As an intriguing fact, one can observe precessing behavior of the electric-dipole polarization inside a ferrite disk.
📄 Content
The concept of nonintegrable, i.e. path-dependent, phase factors is one of the fundamental concepts of electromagnetism. When there is no symmetry with rotational and/or translational invariance and so the wave equation cannot be separated in some coordinate system, one has an example of a nonintegrable system. Presently, nonintegrable systems (such, for example, as Sinai billiards) are the subject for intensive studies in microwave cavity experiments [1]. In view of the so-called quantum-classical correspondence, these experiments are useful in studying the quantum chaos phenomena. To get microwave billiards with broken time-reversal symmetry, ferrite samples were introduced into the resonators [2]. In our recent paper [3], we studied the microwave vortices in a three dimensional system of a TE-polarized rectangular-waveguide resonator with an inserted thin ferrite disk based on full Maxwell-equation numerical solutions of the problem. Because of inserting a piece of a magnetized ferrite into the resonator domain, a microwave resonator behaves under odd time-reversal symmetry (TRS) and a ferrite disk acts as a topological defect causing induced vortices. The microwave vortices are defined as lines to which the Poynting vector is tangential. The interplay of reflection and transmission at the disk interfaces together with material gyrotropy effect gives rise to a rich variety of wave phenomena. It was shown that the power-flow lines of the microwave-cavity field interacting with a ferrite disk, in the proximity of its ferromagnetic resonance, form the whirlpool-like electromagnetic vortices.
In different studies with TE polarized cavities [1 -3], the vortex behavior in a vacuum region of a cavity can be easily understood from an analysis of the field structure. For TE polarized (with an electric field directed along y axis) electromagnetic waves in vacuum, the singular features of the complex electric field component ) , ( z x E y can be related to those that will subsequently appear in . As it was shown in [3], specifically the cases when a ferrite disk is placed in a maximum of the cavity electric field show the most pronounced and compact Poynting-vector vortices.
In the standard situation of microwave cavity experiments with inserted ferrite samples, the mechanism behind the TRS breaking effects is believed to be intimately connected with (and in fact generated by) the losses of energy and flux. Interplay between losses and quantum chaotic effects is rather interesting and non-trivial (see e.g. [4]). At the same time, it is known that the losses in the ferrite are always one of the main problems to study quantum manifestations of classical chaos in the regime of broken TRS. This fact makes a general comparison between the theory and experiment not an easy task. For this reason, any experimental technique able to generate TRS-breaking effects with introducing minimal losses (or introducing them in a controllable way) is of considerable interest.
In the ferromagnetic resonance, depending on a quantity of a bias magnetic field and (or) frequency, one has the regions with positive or negative permeability parameters [5]. For a ferrite disk placed in a maximum of the RF electric field in a rectangular-waveguide resonator, there are fundamentally different conditions for generation of microwave vortices in the positive-and negative-parameter regions. For negative permeability parameters, microwave vortices appear only when the material properties of a disk are characterized by big losses [6]. Similar situation takes place for a plasmon-resonance nanoparticle illuminated by the electromagnetic field, where the spiral energy flow line trajectories appear for a lossy sample with negative permittivity parameters [7]. Contrary, in a case of positive permeability parameters, the losses may play an indirect role in forming microwave vortices. This fact was illustrated in [3] (see Figs. 15 in [3]): in a case of positive permeability parameters one has very slight variation of the Poynting-vector vortex pictures in (and in close vicinity of) a ferrite disk for different losses parameters of a ferrite material.
The purpose of this letter is to show that for a ferrite disk with positive permeability parameters, creation of microwave vortices in a cavity is due to effective chiral magnetic currents rather than due to additional losses (which can be indeed minimal). As a very intriguing fact one can observe topological magnetic charges and precessing electric polarization in microwave ferrite samples, which appear because of the geometrical factor and the TRS breaking. Since the nonintegrable nature of the problem precludes exact analytical results for the eigenvalues and eigenfunctions, numerical approaches are required. We used the HFSS (the software based on FEM method produced by ANSOFT Company) CAD simulation programs for 3D numerical modeling of Maxwell equations [8]. In our numerical experiments, both modulus a
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